Efficient encoding methods

  

Coding theory

refers to study of code properties and their suitability to specific applications. Efficient codes are used, e.g., in data compression, cryptography, error-correction, and group testing.

Codes play a central part in information theory, in particular in the design of efficient and reliable data transmission methods.  Encoding methods focus on reduction (clever use) of redundancy in data compression (in error detection and correction mechanisms) 26/04/2020 Applied Algorithmics - week6 1

Data compression

  

Data compression

is the process of encoding information using fewer bits or other information-bearing units.

Compression is possible where the input data have

statistical redundancy

(e.g., in text files) or when relatively minor changes leading to smaller representation do not affect the

quality/fidelity

of the input (e.g., in pictures, video, or audio files).

Popular instances of data compression that many computer users are familiar with is the ZIP file format (texts), jpeg format (pictures) and mpeg format (for audio and video).

26/04/2020 Applied Algorithmics - week6 2

Data compression

  Some compression schemes are

reversible compression

).

so that the original data can be reconstructed (

lossless data compression

), while others accept some loss of data in order to achieve higher compression (

lossy data

Compression is important because it helps reduce the consumption of expensive resources, such as disk space or connection bandwidth. However, compression requires increased information processing power, which can also be expensive. 26/04/2020 Applied Algorithmics - week6 3

Data compression - simple example

    

Run-Length Encoding

Data files frequently contain the same character repeated many times in a row. For example, text files use multiple spaces to separate sentences, indent paragraphs, format tables & charts, etc. Digitized signals can also have runs of the same value, indicating that the signal is not changing. For example, an image of the night-time sky would contain long runs of the character or characters representing the black background.

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Data compression - simple example



Run-Length Encoding

 In this scheme we focus on long runs of characters.

   Each time a long run is encountered in the input data, two values are written to the output file. The first of these values is the character itself, i.e., a flag to indicate that run-length compression is beginning. The second value is the number of characters in the run.

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Move to Front Transform

 

Move to Front

(

MTF

) transform is an encoding of data (typically a stream of bytes) designed to improve the performance of

entropy encoding

(coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols) techniques of compression. When properly implemented, it is fast enough that its benefits usually justify including it as an extra step in data compression algorithms. 26/04/2020 Applied Algorithmics - week6 6

Move to Front Transform

   In the context of

MTF

each byte value is encoded by its index in a list, which changes over the course of the algorithm. The list is initially stored, e.g., in order by byte value (0, 1, 2, 3, ..., 255). Therefore, the first byte is always encoded by its own value. However, after encoding a byte, that value is moved to the front of the list before continuing to the next byte.

26/04/2020 Applied Algorithmics - week6 7

Move to Front Transform - example

  Let S=<9,9,8,8,8,1,9,9,9> be an input sequence and the initial content of the queue Q is [0,1,2,3,4,5,6,7,8,9] The encoding process will transform S as follows:            S=< 9 ,9,8,8,8,1,9,9,9> and Q=[0,1,2,3,4,5,6,7,8, 9 ] S=< 9 , 9 ,8,8,8,1,9,9,9> and Q=[ 9 ,0,1,2,3,4,5,6,7,8] S=< 9 , 0 , 8 ,8,8,1,9,9,9> and Q=[9,0,1,2,3,4,5,6,7, 8 ] S=< 9 , 0 , 9 , 8 ,8,1,9,9,9> and Q=[ 8 ,9,0,1,2,3,4,5,6,7] S=< 9 , 0 , 9 , 0 , 8 ,1,9,9,9> and Q=[ 8 ,9,0,1,2,3,4,5,6,7] S=< 9 , 0 , 9 , 0 , 0 , 1 ,9,9,9> and Q=[8,9,0, 1 ,2,3,4,5,6,7] S=< 9 , 0 , 9 , 0 , 0 , 3 , 9 ,9,9> and Q=[1,8, 9 ,0,2,3,4,5,6,7] S=< 9 , 0 , 9 , 0 , 0 , 3 , 2 , 9 ,9> and Q=[ 9 ,1,8,0,2,3,4,5,6,7] S=< 9 , 0 , 9 , 0 , 0 , 3 , 2 , 0 , 9 > and Q=[ 9 ,1,8,0,2,3,4,5,6,7] S=< 9 , 0 , 9 , 0 , 0 , 3 , 2 , 0 , 0 > and Q=[ 9 ,1,8,0,2,3,4,5,6,7] Where the blue value refers to the position of the symbol in the last instance of Q 26/04/2020 Applied Algorithmics - week6 8

Burrows-Wheeler Transform

    

The Burrows-Wheeler transform

(

BWT

), a.k.a. block-sorting compression, is one of the most popular method in data compression. It was invented by Michael Burrows and David Wheeler, in 90-ties.

When a character string is transformed by the BWT, none of its characters change value. The transform rearranges in clever for the order of the characters in the string. If the original string had several substrings that occurred frequently, then the transformed string will have several places where a single character is repeated multiple times in a row. This is useful for compression, since it tends to be easy to compress a string that has runs of repeated characters by techniques such as

move-to-front transform

and

run-length encoding

.

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Cyclic rotations

 For

0

≤

k

≤

n-1

, the

k th

cyclic rotation of the string

w

=

w[0..n-1]

is another string

v

=

v[0..n-1]

,

s.t.,

v[i]

=

w[(i+k) mod n]

w v

k

x y y x

26/04/2020 Applied Algorithmics - week6 10

Burrows-Wheeler Transform

 The

Burrows-Wheeler Transform

transform for the string

w

=

w[0..n-1]

is defined as follows:  Create a square matrix

M[n x n]

contains the

k th

in which the cyclic rotation of

w k th

row    Sort rows of

M

in lexicographic order Store the string represented by the last column of

M

And the index of row which contains the position of the original string

w

(i.e.,

0 th

cyclic rotation) 26/04/2020 Applied Algorithmics - week6 11

Burrows-Wheeler Transform (example)

 Consider

5

th Fibonacci word

f 5

=

babbabab

[0] [1] [2] [3] [4] [5] [6] [7]

b a b b a b a b a b b a b a b b b b a b a b b a b a b a b b a b a b a b b a b b b a b b a b b a a b b a b b a b b b a b b a b a

BWT

[0] [1] [2] [3] [4] [5] [6] [7]

a b a b b a b b a b b a b a b b a b b a b b a b b a b a b b a b b a b b a b a b b a b b a b b a b b a b a b b a b b a b b a b a

[4] [1] [6] [3] [0] [5] [2] [7]  26/04/2020 The output string

bbbbbaaa

and position [4] Applied Algorithmics - week6 12

Burrows-Wheeler Transform

 The Burrows-Wheeler transform can be computed by the algorithm that constructs suffix arrays  Which means that the Burrows-Wheeler transform can be computed

in linear time

 The Burrows-Wheeler transform is reversible and the original string can be recovered efficiently via generation of consecutive columns of matrix

M

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a a a b b b b b

Burrows-Wheeler (reverse) Transform --

……..

……..

……..

……..

……..

……..

……..

……..

hard way

b b b b b a a a ba ba ba bb bb ab ab ab ab ab ba ba ba bb bb ab ……..

……..

……..

……..

……..

……..

……..

…….

b b b b b a a a bab bab bab bba bba aba abb abb aba ……..

abb abb bab bab bab bba bba ……..

……..

……..

……..

……..

……..

…….

b b b b b a a a baba babb babb bbab bbab abab abba abba abab ……..

abba abba baba babb babb bbab bbab ……..

……..

……..

……..

……..

……..

…….

b b b b b a a a babab babba babba bbaba bbabb ababb abbab abbab ababb …..

abbab abbab babab babba babba bbaba bbabb …..

…..

…..

…..

…..

…..

…..

b b b b b a a a

26/04/2020

bababb babbab babbab bbabab bbabba ababba abbaba abbabb ababba …..

abbaba abbabb bababb babbab babbab bbabab bbabba …..

…..

…..

…..

…..

…..

…..

b b b b b a a a bababba babbaba babbabb bbababb bbabbab ababbab abbabab abbabba

Applied Algorithmics - week6

ababbab ..

abbabab ..

abbabba ..

bababba babbaba babbabb bbababb bbabbab ..

..

..

..

..

b b b b b a a a bababbab babbabab babbabba bbababba bbabbaba ababbabb abbababb abbabbab ababbabb abbababb abbabbab bababbab babbabab babbabba bbababba bbabbaba

14

Burrows-Wheeler (reverse) Transform - easy way based on stable sorting property

Corresponding symbols

b b b b b a a a

BWT

b b b b b a a a

1 st col

b b a a a b b b

Structure

b b b b b a a a

Reverse BWT

b b b b b a a a b b b b b a a a

1 st col BWT Just follow the cycle

b a b b a b a b

0 1 2 3 4 5 6 7 4 1 6 3 0 5 2 7

26/04/2020 Applied Algorithmics - week6 15

Lempel-Ziv-Welch Compression

   The

Lempel-Ziv-Welch (LZW) dictionary

compression algorithm is an example of dictionary based methods, in which longer fragments of the input text are replaced by much shorter references to code words stored in the special set called LZW is an implementation of a lossless data compression algorithm developed by Abraham Lempel and Jacob Ziv. It was published by Terry Welch in 1984 as an improved version of the LZ78 dictionary coding algorithm developed by Lempel and Ziv. 26/04/2020 Applied Algorithmics - week7 16

LZW Compression

   The key insight of the method is that it is possible to automatically build a dictionary of previously seen strings in the text being compressed. The dictionary starts off with 256 entries, one for each possible character (single byte string). Every time a string not already in the dictionary is seen, a longer string consisting of that string appended with the single character following it in the text, is stored in the dictionary.

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LZW Compression

   The output consists of integer indices into the dictionary. These initially are 9 bits each, and as the dictionary grows, can increase to up to 16 bits. A special symbol is reserved for "flush the dictionary" which takes the dictionary back to the original 256 entries, and 9 bit indices. This is useful if compressing a text which has variable characteristics, since a dictionary of early material is not of much use later in the text.

This use of variably increasing index sizes is one of Welch's contributions. Another was to specify an efficient data structure to store the dictionary.

26/04/2020 Applied Algorithmics - week7 18

LZW Compression - example

   Fibonacci language: w -1 =a, w -2 =b, w i = w i-1 ·w i-2 for i>1 For example, w 6 = babbababbabba We show how LZW compresses babbababbabba

CW 0 CW 1 CW 2 CW 3 CW 4 CW 5

b a b a b b a b a b b a -2 -1

0 1 Virtual part In general: And in particular:

26/04/2020

2 3 4 5 6 CW 4 = CW 3 o First(CW 5 )

Applied Algorithmics - week7

7 11 12 CW i = CW j o First(CW i+1 ) and j

b

13

b

14

a

15

19

LZW Compression - example

       

cw -2

=

b cw -1

=

a cw 0

=

ba cw 1

=

ab cw 2

=

bb cw 3

=

bab cw 4

=

babb cw 5

=

babba

26/04/2020

cw -1 cw 1 a b a b cw 0 cw 3 b cw -2 b b cw 2

Applied Algorithmics - week7

cw 4 a cw 5

20

LZW Compression - compression stage

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LZW Compression - compression stage

cw

  ;

while

( read next symbol

s

from

IN

)

if cw·s

exists in the dictionary

then cw



cw·s

;

else

add

cw·s

to the dictionary; save the index of

cw cw



s

; in

OUT

; 26/04/2020 Applied Algorithmics - week7 22

Decompression stage

Input IN – Compressed file of integers.

Output OUT – Decompressed file of characters. |IN| = Z – Size of the compressed file.

   Copy all numbers from file

IN

to vector

V [256………..Z+255]

Create vector

F [256………..Z+255]

containing first characters of each code word Create vector

CW [256………..Z+255]

of all code words

for

i

=

256

to

Z+255

do if

V[i]

<

256 CW[i]



then

Concatenate(

char(V[i])

,

F[i+1]

)

else

CW[i]

 Concatenate(

CW(V[i])

,

F[i+1]

)  Write to the output file OUT all code words without their last symbols 26/04/2020 Applied Algorithmics - week7 23

LZW text compression



Theorem:

For any input string

S

LZW algorithm computes its compressed counterpart in time

O(n)

, where

n

is the length of

S

.

Sketch of proof: The most complex operations are performed on dictionary. With a help of hash tables all operations can be performed in linear time.

 Also the decompression stage is linear.

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